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Subsystems of the Schauder system that are quasibases for $L^p[0,1], 1 \le p < +\infty$

Authors: K. S. Kazarian and Robert E. Zink
Journal: Proc. Amer. Math. Soc. 126 (1998), 2883-2893
MSC (1991): Primary 46B15, 42C15, 42C20; Secondary 41A30, 41A58
MathSciNet review: 1452807
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Abstract | References | Similar Articles | Additional Information

Abstract: We show that if $\Phi =\{\varphi _i\colon i=1,2,\ldots \}$ is a subsystem of the Faber-Schauder system, and if $\Phi$ is complete in $L^2[0,1]$, then $\Phi$ is a quasibasis for each space $L^p[0,1]$, $1\le p<+\infty$. Although it follows from the work of Ul’yanov that each element of $L^p[0,1]$ can be represented by a Schauder series that converges unconditionally to the function, in the metric of the space, it proves to be the case that none of the aforementioned systems is an unconditional quasibasis for any of the $L^p$-spaces herein considered.

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Additional Information

K. S. Kazarian
Affiliation: Department of Mathematics, Universidad Autonoma de Madrid, 28049 Madrid, Spain; Mathematical Science Division, Purdue University, West Lafayette, Indiana 47907-1968

Robert E. Zink
Affiliation: Department of Mathematics, Universidad Autonoma de Madrid, 28049 Madrid, Spain

Received by editor(s): February 16, 1996
Received by editor(s) in revised form: February 21, 1997
Additional Notes: The first author was supported, in part, by Grant PB94/149 from DGICYT and also by Grant MVR000 from the I.S.F
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1998 American Mathematical Society